The results quoted above lie at the heart of the strategy for the proportionate growth asymptotic carried out by Benaych-Georges and Rao Nadakuditi [1]}, [2]}. That strategy is able to employ convenient convergence arguments. Indeed, as \(\beta _n \rightarrow \beta > 0\) , almost surely, the ESD of noise eigenvalues \(F_n\)  converges weakly to \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}F\hspace{-0.83328pt}}\hspace{0.83328pt}_\beta \) , the Marchenko-Pastur law with parameter \(\beta \) . Moreover, the top noise eigenvalue \(\lambda _1 \xrightarrow{} (1+\sqrt{\beta })^2\) , the upper bulk edge, i.e. the upper endpoint of the support of \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}F\hspace{-0.83328pt}}\hspace{0.83328pt}_\beta \) . Let \([a,b]\)  be a compact interval outside the bulk, having \((1+\sqrt{\beta })^2 < a\) . By the Arzela-Ascoli theorem, the empirical Stieltjes transform of the noise eigenvalues \(s_n(z)\)  converges uniformly on \([a,b]\)  to \(\bar{s}_\beta (z)\) , the Stieltjes transform of \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}F\hspace{-0.83328pt}}\hspace{0.83328pt}_\beta \) . The implications of this key observation include the uniform convergence of \(M_n(z)\)  to a non-random matrix \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_\beta (z)\)  (the fixed-\(\beta \)  analog of \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_n(z)\) ). From the uniform convergence of \(M_n(z)\) , the convergence of the roots of \(\det M_n(z)\)  to those of \(\det \hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_\beta (z)\)  may be established.
